| L(s) = 1 | − 1.56·2-s + 0.434·4-s + 1.69·5-s − 7-s + 2.44·8-s − 2.64·10-s + 1.57·11-s + 5.37·13-s + 1.56·14-s − 4.68·16-s − 7.38·17-s − 0.683·19-s + 0.737·20-s − 2.46·22-s + 8.53·23-s − 2.11·25-s − 8.38·26-s − 0.434·28-s − 5.81·29-s + 7.97·31-s + 2.41·32-s + 11.5·34-s − 1.69·35-s − 11.4·37-s + 1.06·38-s + 4.14·40-s − 5.44·41-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.217·4-s + 0.759·5-s − 0.377·7-s + 0.863·8-s − 0.837·10-s + 0.475·11-s + 1.49·13-s + 0.416·14-s − 1.17·16-s − 1.79·17-s − 0.156·19-s + 0.164·20-s − 0.524·22-s + 1.77·23-s − 0.423·25-s − 1.64·26-s − 0.0821·28-s − 1.08·29-s + 1.43·31-s + 0.427·32-s + 1.97·34-s − 0.287·35-s − 1.87·37-s + 0.173·38-s + 0.655·40-s − 0.849·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 - 1.69T + 5T^{2} \) |
| 11 | \( 1 - 1.57T + 11T^{2} \) |
| 13 | \( 1 - 5.37T + 13T^{2} \) |
| 17 | \( 1 + 7.38T + 17T^{2} \) |
| 19 | \( 1 + 0.683T + 19T^{2} \) |
| 23 | \( 1 - 8.53T + 23T^{2} \) |
| 29 | \( 1 + 5.81T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 - 4.36T + 43T^{2} \) |
| 47 | \( 1 - 3.11T + 47T^{2} \) |
| 53 | \( 1 + 7.03T + 53T^{2} \) |
| 59 | \( 1 - 5.82T + 59T^{2} \) |
| 61 | \( 1 + 6.57T + 61T^{2} \) |
| 67 | \( 1 + 6.68T + 67T^{2} \) |
| 71 | \( 1 + 4.90T + 71T^{2} \) |
| 73 | \( 1 + 7.91T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 6.39T + 83T^{2} \) |
| 89 | \( 1 + 2.66T + 89T^{2} \) |
| 97 | \( 1 - 5.37T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.53550310675468913213219939302, −6.78720979801600672941958934112, −6.38955220944490650361070033159, −5.53822847508634122812984228637, −4.62227382912544251626944405556, −3.93160809650420639610281645816, −2.92550387835020930174111774169, −1.84309822709592688985233697409, −1.24810865734421170113780831839, 0,
1.24810865734421170113780831839, 1.84309822709592688985233697409, 2.92550387835020930174111774169, 3.93160809650420639610281645816, 4.62227382912544251626944405556, 5.53822847508634122812984228637, 6.38955220944490650361070033159, 6.78720979801600672941958934112, 7.53550310675468913213219939302
